Is the harmonic function constant?

Question

Suppose $f$ is harmonic on $\mathbb{R^{2}}$ and constant on a neighbourhood in $\mathbb{R^{2}}$. Is $f$ constant on $\mathbb{R^{2}}$?

Answer 1

Yes. In the plane this is very easy: First, we can assume $f$ is real-valued. Now there exists an entire function $F$ with $f=\Re F$. The Open Mapping Theorem (or various other things) shows that $F$ is constant on that open set where $f$ is constant; hence $F$ is constant in $\Bbb C$.

The same result holds for harmonic functions in $\Bbb R^n$, although it's not as obvious. For example, one could use the Poisson integral formula in balls to show that a harmonic function is real-analytic.